Topology of tropical moduli spaces of weighted stable curves in higher genus
Abstract
Given integers $g \geq 0$, $n \geq 1$, and a vector $w \in (\mathbb{Q} \cap (0, 1])^n$ such that ${2g - 2 + \sum w_i > 0}$, we study the topology of the moduli space $\Delta_{g, w}$ of $w$-stable tropical curves of genus $g$ with volume 1. The space $\Delta_{g, w}$ is the dual complex of the divisor of singular curves in Hassett's moduli space of $w$-stable genus $g$ curves $\overline{\mathcal{M}}_{g, w}$. When $g \geq 1$, we show that $\Delta_{g, w}$ is simply connected for all values of $w$. We also give a formula for the Euler characteristic of $\Delta_{g, w}$ in terms of the combinatorics of $w$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2020
- DOI:
- 10.48550/arXiv.2010.11767
- arXiv:
- arXiv:2010.11767
- Bibcode:
- 2020arXiv201011767K
- Keywords:
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- Mathematics - Combinatorics;
- Mathematics - Algebraic Geometry;
- 14T05
- E-Print:
- 14 pages